A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C.


Introduction
For algebraic vector bundles with an appropriate orientation, there are Euler classes and numbers enriched in bilinear forms. We will start over a field k, and then discuss more general base schemes, obtaining integrality results. Let GW(k) denote the Grothendieck-Witt group of k, defined to be the group completion of the semi-ring of non-degenerate, symmetric, k-valued, bilinear forms, see e.g. [Lam05]. Let a in GW(k) denote the class of the rank 1 bilinear form (x, y) → axy for a in k * .
For a smooth, proper k-scheme f : X → Spec k of dimension n, coherent duality defines a trace map η f : H n (X, ω X/k ) → k, which can be used to construct the following Euler number in GW(k). Let V → X be a rank n vector bundle equipped with a relative orientation, meaning a line bundle L on X and an isomorphism ρ : det V ⊗ ω X/k → L ⊗2 . For 0 ≤ i, j ≤ n, let β i,j denote the perfect pairing (1) β i,j : H i (X, ∧ j V * ⊗ L) ⊗ H n−i (X, ∧ n−j V * ⊗ L) → k given by the composition For i = n−i and j = n−j, note that β i,j is a bilinear form on H i (X, ∧ j V * ⊗L). Otherwise, β i,j ⊕β n−i,n−j determines the bilinear form on H i (X, ∧ j V * ⊗ L) ⊕ H n−i (X, ∧ n−j V * ⊗ L). The alternating sum ind PH x σ.
The index ind PH x σ can be computed explicitly with a formula of Scheja-Storch [SS75] or Eisenbud-Levine/Khimshiashvili [EL77] [Khi77] (see Sections 2.4 and 2.3) and is also a local degree [KW19] [BBM + 21] (this is discussed further in Section 7). For example, when x is a simple zero of σ with k(x) = k, the index is given by a well-defined Jacobian Jac σ of σ, ind PH x σ = Jac σ(x) , illustrating the relation with the Poincaré-Hopf formula for topological vector bundles. (For the definition of the Jacobian, see the beginning of Section 6.2.) In [KW21, Section 4, Corollary 36], it was shown that n PH (V, σ) = n PH (V, σ ′ ) when σ and σ ′ are in a family over A 1 L of sections with only isolated zeros, where L is a field extension with [L : k] odd. We strengthen this result by equating n PH (V, σ) and n GS (V ); this is the main result of §2.
Theorem 1.1 (see §2.4). Let k be a field, and V → X be a relatively oriented, rank n vector bundle on a smooth, proper k-scheme of dimension n. Suppose V has a section σ with only isolated zeros. Then n PH (V, σ) = n GS (V ).
In particular, n PH (V, σ) is independent of the choice of σ.
1.1. Sketch proof and generalizations. The proof of the above theorem proceeds in three steps.
(0) For a section σ of V , we define an Euler number relative to the section using coherent duality and denote it by n GS (V, σ, ρ). If σ = 0, we recover the absolute Euler number n GS (V, ρ), essentially by construction. (1) For two sections σ 1 , σ 2 , we show that n GS (V, σ 1 , ρ) = n GS (V, σ 2 , ρ). To prove this, one can use homotopy invariance of Hermitian K-theory, or show that n GS (V, σ 1 , ρ) = n GS (V, ρ) by showing an instance of the principle that alternating sums, like Euler characteristics, are unchanged by passing to the homology of a complex.
Regular sequences and immersions. Following e.g. [BGI + 71], by a regular immersion of schemes we mean what is called a Koszul-regular immersion in [Sta18, Tag 0638], i.e. a morphism which is locally a closed immersion cut out by a Koszul-regular sequence. Moreover, by a regular sequence we will always mean a Koszu-regular sequence [Sta18,Tag 062D], and we reserve the term strongly regular sequence for the usual notion. A strongly regular sequence is regular [Sta18, Tag 062F], whence a strongly regular immersion is regular. In locally noetherian situations, regular immersions are strongly regular [Sta18, Tags 063L].
Cotangent complexes. For a morphism f : X → Y , we write L f for the cotangent complex. Recall that if f is smooth then L f ≃ Ω f , whereas if f is a regular immersion then L f ≃ C f [1], where C f denotes the conormal bundle.
Graded determinants. We write det : K(X) → P ic(D(X)) for the determinant morphism from Thomason-Trobaugh K-theory to the groupoid of graded line bundles. If C is a perfect complex, then we write detC for the determinant of the associated K-theory point. We write det C ∈ Pic(X) for the ungraded determinant.
Given an lci morphism f , we put ω f = det L f and ω f = detL f . We systematically use graded determinants throughout the text. For example we have the following compact definition of a relative orientation.
Definition 1.5. Let π : X → S be an lci morphism and V a vector bundle on X. By a relative orientation of V /X/S we mean a choice of line bundle L on X and an isomorphism ρ : Hom( detV * , ω X/S ) Note that if π is smooth, this just means that the locally constant functions x → rank(V x ) and x → dim π −1 (π(x)) on X agree, and that we are given an isomorphism L ⊗2 ≃ ω X/S ⊗ det V . Hence we recover the definition from [KW21, Definition 17].

Equality of coherent duality and Poincaré-Hopf Euler numbers
We prove Theorem 1.1 in this section.
The derived category D(k) is equivalent to the category of graded k-vector spaces, by taking cohomology 3 . If V is a (non-degenerate) symmetric bilinear form in graded k-vector spaces, denote by V (n) = V n ⊕ V −n (for n = 0) and V (0) = V 0 the indicated subspaces; observe that they also carry (non-degenerate) symmetric bilinear forms. Definition 2.1. For a relatively oriented rank n vector bundle V → X with section σ and orientation ρ, over a smooth and proper variety f : X → k of dimension n, the Grothendieck-Serre-duality Euler number with respect to σ is n GS (V, σ, ρ) = i≥0 (−1) i [(Rf * β (V,σ,ρ) ) (i) ] ∈ GW(k).
Remark 2.2. In order to not clutter notation unnecessarily, we also write the above definition as We shall commit to this kind of abuse of notation from now on.
Recall that n GS (V, ρ) ∈ GW(k) was defined in the introduction, in terms of the symmetric bilinear form on i,j H i (X, Λ j V * ⊗ L). Proposition 2.3. For any section σ we have n GS (V, σ, ρ) = n GS (V, ρ) ∈ GW(k).
To prove Proposition 2.3, we use the hypercohomology spectral sequence E i,j r (K • ) associated to a complex K • of locally free sheaves on X with Given a perfect symmetric pairing of chain complexes β : The following properties hold: (1) Placing the k in the codomain of β 1 in bidegree (−n, n), β 1 is a map of bigraded vector spaces and satisfies the Leibniz rule with respect to d 1 . It thus induces β 2 : E * , * 2 (K • ) ⊗ E * , * 2 (K • ) → k. Then β 2 satisfies the Leibnitz rule with respect to d 2 and hence induces β 3 , and so on.
(2) All the pairings β i are perfect.
(3) The pairing β ′ is compatible with the filtration in the sense that β ′ (F i , F k ) = 0 if i + k > −n.
(4) It follows that β ′ induces a pairing on gr • R * f * K • . Under the isomorphism gr • ≃ E ∞ , it coincides with β ∞ . (5) β ′ is perfect in the filtered sense: the induced pairing F i ⊗ R * f * K • /F −n−i+1 → k is perfect. (In particular the pairing β ′ is perfect.) Remark 2.4. We do not know a reference for these facts, and proving them would take us too far afield.
The main idea is that we have a sequence of duality-preserving functors Here C perf (X) denotes the category of bounded chain complexes of vector bundles, D(X) fil is the filtered derived category [GP18], and σ • is the "stupid truncation" functor (composed with forgetting to the filtered derived category). The first duality is with respect to Hom(−, ω[n]), the second with respect to Hom(−, σ • (ω[n])) = Hom(−, ω[n](−n)), and the third with respect to Hom(−, k[0](−n)). There are further duality preserving functors where D(X) gr = Fun(Z, D(X)), with Z viewed as a discrete category. Hence any perfect pairing C ⊗C → k[0](−n) ∈ D(k) fil induces a perfect pairing on H * C gr ⊗ H * C gr → k(−n, n), satisfying (1), and a pairing H * U C ⊗ H * U C → k, satisfying (3, 5). Moreover there is a spectral sequence E 1 = H * C gr ⇒ H * U C, 3 In this section, we treat all categories as 1-categories, i.e. ignore the higher structure of D(k) as an ∞-category.
satisfying (1) and (4). (2) is obtained from the fact that passage to homology is a duality preserving functor. We apply this to K • ∈ C perf (X); then gr i σ Lemma 2.5. Let X be a graded k-vector space with a finite decreasing filtration Suppose X ⊗ X → k is a perfect symmetric bilinear pairing, which is compatible with the filtration in the sense of (3) and (5). Let X i denote the ith graded subspace of X and X i • denote the ith graded subspace of X • . Then in GW(k), there is equality Proof. Note that (5) implies that the pairing gr • X is non-degenerate, so the statement makes sense (recall Remark 2.2). On any graded symmetric bilinear form, the degree i and −i part for i = 0 assemble into a metabolic space, with Grothendieck-Witt class determined by the rank (see Lemma B.2). It is clear that the ranks on both sides of our equation are the same; hence it suffices to prove the Lemma in the case where X i = 0 for i = 0. We may thus ignore the gradings.
Let N be maximal with the property that X N = 0. We have a perfect pairing Since X N +1 = 0 we deduce that X −n−N = X and hence X j = X for all j ≤ −n − N . If −n − N ≥ N then X = X N (N ) and there is nothing to prove; hence assume the opposite. We have the perfect pairing Extend the filtration X ′ by zero on the left and constantly on the right. By construction, X ′gr i = X gr i for i = N, −n − N , and the pairing on X ′ ⊂ X is perfect in the filtered sense. By [MH73,Lemma I This holds since both sides are metabolic of the same rank: X −n−N is an isotropic subspace of half rank on either side (see again Lemma B.2). Lemma 2.6. Let E • be a chain complex with a non-degenerate, symmetric bilinear form Proof. Since passing to homology is a duality preserving functor, the statement makes sense. Both sides have the same rank, so it suffices to prove equality in W(k) (see Lemma B.2). We have a perfect pairing C i ⊗ C −i → k and similarly for homology. Both are metabolic unless i = 0. We can choose a splitting here H ⊂ ker(C 0 → C 1 ) maps isomorphically to H 0 (C). The restriction of the pairing on C 0 to H is perfect by construction, and hence C 0 = H ⊕H ⊥ . It suffices to show that H ⊥ is metabolic. Compatibility of the pairing with the differential shows that d(C −1 ) ⊂ C 0 is an isotropic subspace. Self-duality shows that im(d : This is the desired result.
Remark 2.7. Admitting a version of Hermitian K-theory which is A 1 -invariant on regular schemes and has proper pushforwards, one can given an alternative proof of Proposition 2.3 by considering the Koszul complex with respect to the section tσ on A 1 × X. While we believe such a theory exists, at the time of writing there is no reference for this in characteristic 2, so we chose to present the above argument instead.

2.2.
Local indices for n GS (V, σ, ρ). Suppose that σ is a section with only isolated zeros. Let i denote the closed immersion i : Z = Z(σ) ֒→ X given by the zero locus of σ. We express n GS (V, σ, ρ) as a sum over the points z of Z of a local index at z. To do this, we use a pushforward in a suitable context and show that β (V,σ) is a pushforward from Z. For a line bundle L on a scheme X, denote by BL naive (D(X), L[n]) the set of isomorphism classes of non-degenerate symmetric bilinear forms on the derived category of perfect complexes on X, with respect to the duality Hom(−, L[n]). For a proper, lci map f : X ′ → X, coherent duality supplies us with a trace map η f,L : f * f ! (L) → L. We can use this to build a pushforward (see [CH09,Theorem 4.2.9]) Example 2.9. Consider the case of a relatively oriented vector bundle V on a smooth proper variety f : X → Spec(k). Note that elements of BL naive (k) are just isomorphism classes of symmetric bilinear forms on graded vector spaces. The orientation supplies us with an equivalence Under the induced pushforward map we have Remark 2.10. A symmetric bilinear form φ on the derived category D(S) is usually not a very sensible notion. We offer three ways around this.
(1) If 1/2 ∈ S, we could look at the image of φ in the Balmer-Witt group of S.
(2) If φ happens to be concentrated in degree zero, it corresponds to a symmetric bilinear form on a vector bundle on S, which is a sensible invariant. (3) If S = Spec(k) is the spectrum of a field, then D(S) is equivalent to the category of graded vector spaces, and we can split φ into components by degree and consider Proposition 2.11. Let X be a scheme, V a vector bundle, and σ ∈ Γ(X, V ) a section locally given by a regular sequence. Write i : Z = Z(σ) ֒→ X for the inclusion of the zero scheme. Proposition B.1 yields a canonical equivalence , where n is the rank of V ; under the induced map is the canonical pairing on the Koszul complex as in §2.1.
Proof. Because σ locally corresponds to a regular sequence, the canonical map r : . We claim that this is the equivalence of Proposition B.1. The proof of said proposition shows that the problem is local on Z, so we may assume that V is trivial. Then this map is precisely the isomorphism constructed in [Har66, Proposition III.7.2 and preceeding pages], which is also the isomorphism employed in the proof of Proposition B.1. Now we prove that i * (1 Z ) = β (V,σ) . Consider the following diagram The map m K : K(V, σ) ⊗ K(V, σ) → K(V, σ) is the canonical multiplication (see (2) §2.1), and m Z : The categorical details are worked out in the reference. Now we get back to our Euler numbers. Let X/k be smooth and proper, V a relatively oriented vector bundle, σ a section of V with only isolated zeros. Write i : Z = Z(σ) ֒→ X for the inclusion of the zero scheme of σ. Let ̟ : Z → Spec k and f : X → Spec k denote the structure maps, so that ̟ = f i.
Since also f ! O k ≃ ω X/k [n] (see e.g. Proposition B.1), we therefore obtain a canonical equivalence We use this equivalence to define Corollary 2.13. With this notation, we have Proof. By Lemma 2.12 we have ̟ * = f * i * . Proposition 2.11 and the projection formula imply that i * (i * L ⊗ i * L → i * L ⊗2 ) = β V,σ,ρ . We conclude by Example 2.9.
Suppose that σ has isolated zeros, or in other words that the support of σ is a disjoint union of points. Then n GS (V, σ, ρ) can be expressed as a sum of local contributions. Namely, for each point z of Z, let i z : Z z ֒→ X denote the chosen immersion coming from the connected component of Z given by z. Let ̟ z : Z z → Spec k denote the structure map. Then In light of this we propose the following.
Definition 2.14. For a relatively oriented vector bundle with a section as above, and z ∈ Z(σ), we define 4 Recall our convention that since we are invoking a functor f ! , Z, Y, X are also finite type and separated over a noetherian base S. Without this we should add the hypothesis that f, g are locally (so globally) of finite presentation.
The above formula then reads In the next two subsections, we compute the local contributions ind z (σ) as an explicit bilinear form constructed by Scheja and Storch [SS75], appearing in the Eisenbud-Levine-Khimshiashvili signature theorem [EL77] [Khi77], and used as the local index of the Euler class constructed in [KW21, Section 4].
2.3. Scheja-Storch and coherent duality. Let S be a scheme, π : X → S a smooth scheme of relative dimension n, and Z ⊂ X closed with ̟ : Z → S finite. Suppose given the following data: (1) Sections T 1 , . . . , T n ∈ O(X) such that T i ⊗ 1 − 1 ⊗ T i generate the ideal of X ⊂ X × S X.
Remark 2.15. Since Z → X is quasi-finite, Lemma B.5 shows that f 1 , . . . , f n is a regular sequence and Z → X is flat, so finite locally free (being finite and finitely presented [Sta18, Tag 02KB]).
Choose a ij ∈ O(X × S X) such that Let ∆ ∈ O(Z × S Z) be the image of the determinant of a ij . Since ̟ is finite locally free, ∆ determines an element∆ of Remark 2.16. We can make∆ explicit: Remark 2.17. By construction, the pullback of ∆ along the diagonal δ : Z → Z × S Z is the determinant of the differentiation map C Z/X → Ω X | Z with respect to the canonical bases. In other words this is the Jacobian: . Theorem 2.18. Under the above assumptions, the map is a symmetric isomorphism and hence determines a symmetric bilinear structure on ̟ * O Z . This is the same structure as ̟ * (1), i.e.
Here the isomorphism with the first isomorphism given by Proposition B.1, and the third isomorphism given by the sections (T i ) and (f i ).
Remark 2.19. The theorem asserts in particular that the isomorphism∆, and hence the section ∆, are independent of the choice of the a ij .
We begin with some preliminary observations before delving into the proof. The problem is local on S, so we may assume that S = Spec(A); then Z = Spec(B). Since ̟ is finite, there is a canonical isomorphism Proof of Theorem 2.18. The isomorphisms determine an element ∆ ′ ∈ Hom A (B * , B). The theorem is equivalent to showing that∆ = ∆ ′ . We thus need to make explicit the isomorphism Tracing through the definitions (including the proof of [Har66, III Proposition 8.2]), one finds that this isomorphism arises by computing Ext n X (B, O X ) in two ways. One the one hand, the kernel of the surjection O X → B is generated by f 1 , . . . , f n , which is a regular sequence by Remark 2.15; let K A (f ) • denote the corresponding Koszul complex. On the other hand we can consider the embedding Z ֒→ X × Z; its ideal is generated by the strongly regular is still a resolution. We shall conflate K B (T − t) and p * K B (T − t) notationally. We can thus compute Letting K A (f ) be the exterior algebra on {e 1 , . . . , e n } and K B (T − t) the exterior algebra on {e ′ 1 , . . . , e ′ n }, the map ζ is specified by ζ(e i ) = jā ij e ′ j . The isomorphism is thus given by Tracing through the definitions, we find that the above composite sends α ∈ Hom A (B, A) to k α(b k )b ′ k . By Remark 2.16, this is precisely∆. This concludes the proof. This form was first constructed, without explicitly using coherent duality, by Scheja and Storch [SS75,3].
Example 2.21. Suppose that Z → S is an isomorphism (where Z = Z(F ) as above), so that the diagonal δ : Z → Z × S Z is also an isomorphism. Then −|− SS is just the rank 1 bilinear form corresponding to multiplication by δ * (∆) ∈ O Z ≃ O S . In other words, using Remark 2.17, −|− SS identifies with (x, y) → (Jac F )xy.
2.4. The Poincaré-Hopf Euler number with respect to a section. In this subsection, we recall the Euler class defined in [KW21, Section 4] and prove Theorem 1.1. To distinguish this Euler class from the others under consideration, here we call it the Poincaré-Hopf Euler number, because it is a sum of local indices as in the Poincaré-Hopf theorem for the Euler characteristic of a manifold. It is defined using local coordinates.
Let k be a field, and let X be an n-dimensional smooth k-scheme. Let z be a closed point of X.  Proof. When k is infinite, this follows from [Knu91, Chapter 8. Proposition 3.2.1]. When k(z)/k is separable, for instance when k is finite, this is [KW21, Lemma 18].
As above, let V be a relatively oriented, rank n vector bundle on X. Let σ be a section with only isolated zeros, and let Z ֒→ X denote the closed subscheme given by the zero locus of σ. Let z be a point of Z. The Poincaré-Hopf local index or degree ind PH z σ ∈ GW(k) was defined in [KW21, Definition 30] as follows. Choose a system of Nisnevich coordinates ϕ : U → A n k around z. After possibly shrinking U , the restriction of V to U is trivial and we may choose an isomorphism ψ : V | U → O n U of V . The local trivialization ψ induces a distinguished section of det V (U ). The system of local coordinates ϕ induces a distinguished section of det T X (U ), and we therefore have a distinguished section of (det V ⊗ ω X )(U ). As in [KW21,Definition 19], the local coordinates ̟ and local trivialization ψ are said to be compatible with the relative orientation if the distinguished element is the tensor square of a section of L(U ). By multiplying ψ by a section in O(U ), we may assume this compatibility.
Under ψ, the section σ can be identified with an n-tuple (f 1 , . . . , f n ) of regular functions, ψ(σ) = (f 1 , . . . , f n ) ∈ ⊕ n i=1 O U . Let m denote the maximal ideal of O U corresponding to z. Since z is an isolated zero, there is an integer n such that m n = 0 in O Z,z . For any N , it is possible to choose (g 1 , . . . , g n ) in Proof of Theorem 1.1. By Proposition 2.3 we have n GS (V ) = n GS (V, σ), where the orientation has been suppressed from the notation, but is indeed present. Using Formula (3), it is thus enough to show that ind z (σ) = ind PH z (σ). This follows from Theorem 2.18. One needs to be careful about the trivializations used in defining the various pushforward maps; this is ensured precisely by the condition that the tautological section is a square. The details of this argument are spelled out more carefully in the proof in Proposition 3.13 in the next section.
One can extend the comparison of local degrees ind PH z σ = ind z σ to work over a more general base scheme S. This was done for S = A 1 k with k a field in [KW21, Lemma 33], but in more generality, it is useful to pick the local coordinates using knowledge of both σ and X as follows.
Definition 2.25. Let X be a scheme, V a vector bundle on X and σ a section of V .
(1) We call σ non-degenerate if it locally corresponds to a regular sequence.
(2) Given another scheme S and a morphism π : X → S, we call σ very non-degenerate (with respect to π) if it non-degenerate and the zero locus Z(σ) is finite and locally free over S. Example 2.27. If σ is a non-degenerate section, then precomposition with σ induces an isomorphism Definition 2.28. Let X be a smooth S-scheme, and let V → X be a vector bundle, relatively oriented by ρ. Let σ be a very non-degenerate section of V , and let Z be a closed and open subscheme of the zero locus Z(σ) of σ. By a system of coordinates for (V, X, σ, ρ, Z) we mean an open neighbourhood U of Z in X, anétale map ϕ : U → A n S , a trivialization ψ : V | U ≃ O n U , and a section σ ′ ∈ O n A n S (ϕ(U )), such that the following conditions hold: ( (3) the canonical section of ω X/S ⊗ det V | Z ∼ = L ⊗2 | Z determined by ψ and ϕ corresponds to the square of a section of L| Z . Here for (2) and (3) we used Example 2.27.
Suppose that X is dimension n over S, so that the rank of V is also n. Let Z ⊂ Z(σ) be a clopen component and write ̟ : Z → S for the structure map. The local index generalizes straightforwardly from Definition 2.14: Remark 2.30. Since ̟ is finite locally free, ̟ * preserves vector bundles. In particular, ind Z (σ) ∈ BL naive (S) is a symmetric bilinear form on a vector bundle, as opposed to just on a complex up to homotopy. (See also Remark 2.10.) Hence Definition 2.20 supplies us with a symmetric bilinear form In particular its isomorphism class is independent of the choice of coordinates.
Proof. The argument is the same as in the proofs of Theorem 1.1 and Proposition 3.13.
In contrast, our proof that the Euler number (sum of indices) is independent of the choice of section (i.e. Proposition 2.3) does not generalize immediately; in fact this will not hold in BL naive (S) but rather in some quotient (like GW(k) in the case of fields). As indicated in Remark 2.7, one situation in which it is easy to see this independence is if the quotient group satisfies homotopy invariance. This suggests studying Euler numbers valued in more general homotopy invariant cohomology theories for algebraic varieties, which is what the remainder of this work is concerned with.

Cohomology theories for schemes
3.1. Introduction. In order to generalize the results from the previous sections, we find it useful to introduce the concept of a cohomology theory twisted by K-theory. We do not seek here to axiomatize all the relevant data, but just introduce a common language for similar phenomena.
Definition 3.1. Let S be a scheme and C ⊂ Sch S a category of schemes. Denote by C L the category of pairs (X, ξ) where X ∈ C and ξ ∈ K(X) (i.e. a point in the K-theory space of X), and morphisms those maps of schemes compatible with the K-theory points. 5 By a cohomology theory E over S (for schemes in C) we mean a presheaf of sets on C L , i.e. a functor To illustrate the flavor of cohomology theory we have in mind, we begin with two examples.
Warning 3.3. For cohomology theories with values in a 1-category (like sets), in the above definition we can safely replace K(X) by its trunctation K(X) ≤1 , i.e. the ordinary 1-groupoid of virtual vector bundles. However, we can in general not replace it by just the set K 0 (X). In other words, if (say) V is a vector bundle on X and φ an automorphism of V , then there is an induced automorphism which may or may not be trivial. For example, in the case E = GW as above, if V = O is trivial and φ corresponds to a ∈ O × (X), then E(φ) is given by multiplication by a ∈ GW(X).

3.2.
Features of cohomology theories. Many cohomology theories that occur in practice satisfy additional properties beyond the basic ones of the above definition, and many come with more data. We list here some of those relevant to the current paper.
Morphisms of theories: Cohomology theories form a category in an evident way, with morphisms given by natural transformations. Trivial bundles: We usually abbreviate E O n (X) to E n (X). Additive and multiplicative structure: Often, E takes values in abelian groups. Moreover, often E 0 (X) is a ring and E ξ (X) is a module over E 0 (X). Typically all of this structure is preserved by the pullback maps. Disjoint unions: Usually E converts finite disjoint unions into products, i.e. E(∅) = * and . If E takes values in abelian groups, this is usually written as . Orientations: In many cases, the cohomology theory E factors through a quotient of the category C L , built using a quotient q : K(X) → K ′ (X) of the K-theory groupoid. In other words, one has canonical isomorphisms E ξ (X) ≃ E ξ ′ (X) for certain K-theory points ξ, ξ ′ . More specifically: GL-orientations: In the above situation, if K ′ (X) = Z and q is the rank map, then we speak of a GL-orientation. In other words, in this situation we canonically have E ξ (X) ≃ E rk(ξ) (X). In particular, Warning 3.3 does not apply: all automorphisms of vector bundles act trivially on E. This happens for example if E = CH (see Example 3.2(1)). SL-orientations: If instead K ′ (X) = P ic(D(X)) via the determinant, then we speak of an SLorientation. In other words, in this situation E ξ (X) only depends on the rank and (ungraded) determinant of ξ. We write E rk(ξ) (X, det(ξ)) for this common group. This happens for example if E = GW (see Example 3.2(1)). SL c -orientations: This is a further strengthening of the concept of an SL-orientation, where in K ′ (X) = P ic(D(X)) we mod out (in the sense of groupoids) by the squares of line bundles. In other words, if L 1 , L 2 , L 3 are line bundles on X, then any isomorphism Note that in particular then E n (X, L) ≃ E n (X, L * ). This also happens for E = GW, essentially by construction. Supports: Often, for Z ⊂ X closed there is a cohomology with support, denoted E ξ Z (X). It enjoys further functorialities which we do not list in detail here. Transfers: In many theories, for appropriate morphisms p : X → Y and ξ ∈ K(Y ), there exists tw(p, ξ) ∈ K(X) and a transfer map compatible with composition. Typically p is required to be lci, and where L p is the cotangent complex [Ill71]. Furthermore, typically p is required to be proper, or else we need to fix Z ⊂ X closed and proper over Y and obtain p * : E . Finally, usually E takes values in abelian groups and satisfies the disjoint union property, and transfer from a disjoint union is just the sum of the transfers.
(1) We have defined a morphism of cohomology theories as a natural transformation of functors valued in sets. Whether or not such a transformation respects additional structure (abelian group structures, orientations, transfers etc.) must be investigated in each case.
(2) In many cases (in particular in the presence of homotopy invariance), SL-oriented theories are also canonically SL c -oriented. See Proposition 4.19.
3.3. Some cohomology theories. We now introduce a number of cohomology theories that can be used in this context.
Hermitian K-theory GW: This is the theory from Example 3.2 (2). It is SL c -oriented. We believe that it has transfers for (at least) smooth proper morphisms and regular immersions, but we are not aware of a reference for this in adequate generality. If X is regular and 1/2 ∈ X one can use the comparison with KO-theory (see below). Naive derived bilinear forms BL naive : See §2.2.
Cohomology theories represented by motivic spectra: Let SH(S) denote the motivic stable ∞-category. Then any E ∈ SH(S) defines a cohomology theory on Sch S , automatically satisfying many good properties; for example they always have transfers along smooth and proper morphisms, as well as regular immersions. For a lucid introduction, see [EHK + 20a]. We recall some of the main points in §4. Orthogonal K-theory spectrum KO: This spectrum is defined and stable under arbitrary base change if 1/2 ∈ S [PW18, ST15]. Over regular bases, it represents Hermitian K-theory GW; in general it represents a homotopy invariant version. Generalized motivic cohomology HZ: This can be defined as π eff 0 (½); see for example [Bac17]. Over fields (of characteristic not 2) it represents generalized motivic cohomology in the sense of Calmès-Fasel [CF17b,CF17a,BF17]; it is unclear if this theory is useful in this form over more general bases.
3.4. The yoga of Euler numbers. Let E be a cohomology theory.
Definition 3.5. We will say that E has Euler classes if, for each scheme X over S and each vector bundle V on X we are supplied with a class Remark 3.6. The twist by V * (instead of V ) in the above definition may seem peculiar. This ultimately comes from our choice of covariant (instead of contravariant) equivalence between locally free sheaves and vector bundles, whereas a contravariant equivalence is used in the motivic Thom spectrum functor and hence in the definition of twists.
Typically, the Euler classes will satisfy further properties, such as stability under base change; we do not formalize this here. Now suppose that π : X → S is smooth and proper, V is relatively oriented, E has transfers for smooth proper maps and is SL c -oriented. In this case we have a transfer map Definition 3.7. In the above situation, we call the Euler number of V in E with respect to the relative orientation ρ.
Example 3.8. Let E = GW. We can define a family of Euler classes by here we use the Koszul complex from §2.1. This depends initially on a choice of section, but we shall show that the Grothendieck-Witt class often does not. In any case, here we chose the 0-section for definiteness. Assuming that GW has transfers (of the expected form) in this context, we find that Now let σ be a non-degenerate section of V (in the sense of Definition 2.25) and write i : Z = Z(σ) ֒→ X for the inclusion of the zero-scheme. Thus i is a regular immersion. In this case one has (see Example 2.27) and consequently, if E has pushforwards along regular immersions, there is a transfer map The following result is true in all cases that we know of; but of course it cannot be proved from the weak axioms that we have listed.
Meta-Theorem 3.9. Let σ be a non-degenerate section of a vector bundle V over a scheme X. Let E be a cohomology theory with Euler classes and pushforwards along regular immersions, such that E 0 (S) has a distinguished element 1 (e.g. is a ring). Then where we use the identification above for the pushforward.
Going back to the situation where X is smooth and proper over S, V is relatively oriented and E is SL c -oriented and has transfers along proper lci morphisms, we also have the pushforward More generally, if Z ′ ⊂ Z is a clopen component, then we have a similar transfer originating from E 0 (Z ′ ).
Definition 3.10. For V, σ, X, E as above, for any clopen component Z ′ ⊂ Z we denote by Meta-Corollary 3.11. Let σ be a non-degenerate section of a relatively oriented vector bundle V over π : X → S. Let E be an SL c -oriented cohomology theory with Euler classes and pushforwards along proper lci morphisms, such that Meta-Theorem 3.9 applies. Then Proof. By assumption, transfers are compatible with composition, and additive along disjoint unions. The result follows.
Example 3.12. If S = Spec(k) is the spectrum of a field, then Z is zero-dimensional, and hence decomposes into a finite disjoint union of "fat points". In particular, the Euler number is expressed as a sum of local indices, in bijection with the zeros of our non-degenerate section.
Recall the notion of coordinates from Definition 2.28. The following result states that indices may be computed in local coordinates.
Proposition 3.13. Let E be an SL c -oriented cohomology theory with Euler classes and pushforwards along proper lci morphisms. Let (ψ, ϕ, σ 2 ) be a system of coordinates for (V, X, σ 1 , ρ 1 , Z). Then where ρ 2 is the canonical relative orientation of O n A n /A n .
Proof. Let ̟ : Z → S denote the canonical map. Then both sides are obtained as ̟ * (1), but conceivably the orientations used to define the transfer could be different; we shall show that they are not. In other words, we are given two isomorphisms here L 2 = O, and for i = 2 we implicitly use ϕ and ψ as well. We first check that the two isomorphisms det N Z/X ≃ det V | Z are the same. Indeed V | U ≃ O n via ψ, and up to this trivialization the isomorphism is given by the trivialization of C Z/X by σ i ; these are the same by assumption (2). Now we deal with the second half of the isomorphism. By construction we have an 1 is a tensor square. Unwinding the definitions, this follows from assumption (3).

Cohomology theories represented by motivic spectra
We recall some background material about motivic extraordinary cohomology theories, i.e. theories represented by motivic spectra. We make essentially no claim to originality. 4.1. Aspects of the six functors formalism. We recall some aspects of the six functors formalism for the motivic stable categories SH(−), following the exposition in [EHK + 20a]. 4.1.1. Adjunctions. For every scheme X, we have a symmetric monoidal, stable category SH(X). For every morphism f : X → Y of schemes we have an adjunction If no confusion can arise, we sometimes write E Y := f * E. If f is smooth, there is a further adjunction If f is locally of finite type, then there is the exceptional adjunction There is a natural transformation α : In particular, given composable morphisms f, g of the appropriate type, we have equivalences (f g) * ≃ f * g * , and so on. 4.1.2. Exchange transformations. Suppose given a commutative square of categories Suppose we have a commutative square of schemes Then we have an induced commutative square of categories Passing to the right adjoints, we obtain the exchange transformation Similarly there is Ex * # : g ′ # f ′ * → f * g # (for g smooth; this is in fact an equivalence if (4) is cartesian), and so on. 4.1.3. Exceptional exchange transformation. Given a cartesian square of schemes as in (4), with g (and hence g ′ ) locally of finite type, there is a canonical equivalence Passing to right adjoints, we obtain 4.1.4. Thom transformation. Given a perfect complex E of vector bundles on X, the motivic J-homomorphism provides us with an invertible spectrum Σ E ½ ∈ SH(X). We denote by Σ E : SH(X) → SH(X), E → E ∧ Σ E ½ the associated invertible endofunctor. If E is a vector bundle (concentrated in degree zero), then Σ E ½ is the suspension spectrum on the Thom space E * /E * \ 0. 6 Lemma 4.1. The functor f ! commutes with Thom transforms.
Proof. This follows from the projection formula [CD19, A.5.1(6)] and invertibility of Σ (−) ½: Then the cotangent complex L f is perfect, and there exists a canonical purity transformation
. This assignment forms a cohomology theory in the sense of §3.1. It takes values in abelian groups, has supports, and satisfies the disjoint union property. We shall see that it has transfers for proper lci maps. It need not be orientable in general.
If Z = X, we may omit it from the notation and just write E ξ (X). As before if ξ is a trivial virtual vector bundle of rank n ∈ Z then we also write E n where we have used that i * ≃ i ! and Lemma 4.1. Using the localization sequence j # j * → id → i * i * [Hoy17, Theorem 6.18(4)] to identify i * ½ ≃ X/X \ Z we find that . Remark 4.3. The final expression in the above example only depends on Z ⊂ X as a subset, not subscheme. This also follows directly from the definition, since SH(Z) ≃ SH(Z red ); this is another consequence of localization.
. This just follows from the fact that π * etc. are functors.
Here the Ex * ! and Ex * ! come from the cartesian square , the transformation Σ E is built out of Spec(Sym(E))-which is the vector bundle corresponding to E * , in our convention.
Lemma 4.4. Let f : X ′ → X ∈ Sch X , Z ⊂ X, ξ ∈ K(Z) and α : E → F ∈ SH(S). The following square commutes Proof. This is just an expression of the fact that the exchange transformations used to build f * are indeed natural transformations. 4.2.4. Covariant functoriality in X. Suppose given a commutative square in Sch S (5) where f is smoothable lci, i, k are closed immersions and g is proper. For every ξ ∈ K(Z 2 ), there is the Gysin map where we used Lemma 4.1 to move Σ L f through i ! and Σ ξ through g ! , and also used i Remark 4.5. In [EHK + 20a], the Gysin map is denoted by f ! , to emphasize that it involves the purity transform. We find the notation f * more convenient.
Lemma 4.6. The following hold.
(1) Suppose given a square as in (5) and α : E → F ∈ SH(S). Then the following square commutes Suppose given a square as in (5) and s : Y ′ → Y such that s, f are tor-independent. Let k ′ , f ′ the induced maps, and so on. Then the following square commutes (3) Suppose given a commutative diagram in Sch S as follows where ζ is a natural transformation of endofunctors of SH(S), this is clear.
Here s X : X ′ → X is the canonical map and ξ ′ = (Z ′ 2 → Z 2 ) * ξ. All the unlabelled equivalences arise from moving Thom transforms through (various) pullbacks, compatibility of pullbacks and pushforwards with composition, and equivalences of the form p * ≃ p ! for p proper. (3) Consider the following diagram. Example 4.7. Consider a commutative square as in (5), with X = Y and f = id, so that g : Z 1 ֒→ Z 2 is a closed immersion. Then f * : is the "extension of support" map. In particular taking Z 2 = X as well, we obtain the map E i * ξ Z1 (X) → E ξ (X) "forgetting the support". Lemma 4.6(3) now in particular tells us that given a proper map f : X → Y , a closed immersion i : Z ֒→ X and ξ ∈ K(Y ), the following diagram commutes Lemma 4.8. The multiplicative structure on E-cohomology is compatible with pullback: given Z 1 , Z 2 ⊂ X, ξ, ξ ′ ∈ K(X), f : X ′ → X, the following diagram commutes Immediate from the definitions. 4.3.2. Thom spectra. Let G = (G n ) n be a family of finitely presented S-group schemes, equipped with a morphism of associative algebras G → (GL nk,S ) n (for the Day convolution symmetric monoidal structure on Fun(N, Grp(Sch S ))). Then there is a notion of (stable) vector bundle with structure group G, the associated K-theory space K G (X), and the associated Thom spectrum MG, which is a ring spectrum [BH17, Example 16.22].
Example 4.9. If G n = GL n , then K G (X) = K(X) and MGL is the algebraic cobordism spectrum [BH17,Theorem 16.13]. If G n = SL n (respectively Sp n ), then K G (X) is the K-theory of oriented (respectively symplectic) vector bundles in the usual sense, and MSL (respectively MSp) is the Thom spectrum as defined in [PW10a].
In order to work effectively with MG, one needs to know that it is stable under base change. This is easily seen to be true for MGL, MSL, MSp [BH17, Example 16.23]. We record the following more general result for future reference.
Proposition 4.10. The Thom spectrum MG is stable under base change, provided that each G n is flat and quasi-affine.
Proof. We have a presheaf K G ∈ P(Sch S ) and a map K G → K. For f : X → S ∈ Sch S , denote by K G X ∈ P(Sm X ) and j X : K G X → K| SmX the restrictions. Then by definition MG X = M X (j X ), where M X : P(Sm X ) /K → SH(X) is the motivic Thom spectrum functor [BH17, §16.1]. Let LK G S ∈ P(Sch S ) denote the left Kan extension of K G S . We claim that LK G S → K G is a Nisnevich equivalence. Assuming this, we deduce that f * K G S ≃ (LK G S )| Sm X → K G X is a Nisnevich equivalence. Since M X inverts Nisnevich equivalences [BH17,Proposition 16.9], this implies that f * MG S ≃ MG X , which is the desired result.
To prove the claim, we first note that by [EHK + 20b, Lemma 3.3.9], we may assume S affine, and it suffices to prove that the restriction of K G to Aff S is left Kan extended from smooth affine S-schemes. By definition K G = (Vect G ) gp , where Vect G = n≥0 BG n (here the coproduct is as stacks, i.e. fppf sheaves). The desired result now follows from [EHK + 20b, Proposition A.0.4 and Lemma A.0.5] (noting that the coproduct of stacks is the same as coproduct of Σ-presheaves, and Kan extension preserves Σ-presheaves). Now let ξ ∈ K G (X). Then there is a canonical equivalence [BH17, Example 16.29] We denote by t ξ ∈ MG ξ−|ξ| (X) the class of the map

Oriented ring spectra.
Definition 4.11. Let E ∈ SH(S) be a ring spectrum and G = (G n ) n a family of group schemes as in §4.3.2. By a strong G-orientation of E we mean a ring map MG → E.
Example 4.12. The spectrum KO is strongly SL-oriented; see Corollary A.3.
Note that if E ∈ SH(S) is strongly G-oriented, then there is no reason a priori why E X should be strongly G X -oriented. This is true if MG is stable under base change, so for most reasonable G by Proposition 4.10. We will not talk about strong G-orientations unless MG is stable under base change, so assume this throughout.
Proposition 4.13. Let E be strongly G-oriented and ξ ∈ K G (X).
In particular, E is G-oriented.
Proof. (1) follows from the same statement for MG, where it holds by construction. For the first half of (2), it suffices to show that t ξ (E) is a unit in the Picard-graded homotopy ring of E. This follows from the same statement for MG. The second half of (2) follows. (3) immediately follows from (1).
(2) There is a canonical isomorphism E n Z (X, L 1 ) ≃ E n Z (X, L 2 ), compatible with base change. In particular, the cohomology theory represented by E is SL c -oriented.

Euler classes for representable theories
5.1. Tautological Euler class. Let E ∈ SH(S) be a ring spectrum, X ∈ Sch S and V a vector bundle on X.
Definition 5.1. We denote by e(V ) = e(V, E) ∈ E V * (X) the tautological Euler class of V , defined as the composite Proof. Immediate.
If E is strongly SL-oriented in the sense of §4.3.3, and hence SL c -oriented in the sense of §3.2, then for any relatively oriented vector bundle V over a smooth and proper scheme X/S we obtain an Euler number n(V, ρ, E) ∈ E 0 (S). See §3.4.

Integrally defined Euler numbers.
Corollary 5.3 (Euler numbers are stable under base change). Let E be a strongly SL-oriented cohomology theory and let V be vector bundle V over a smooth and proper scheme X/S, relatively oriented by ρ. Let f : S ′ → S be a morphism of schemes. Then Proof. This holds since all our constructions are stable under base change. See in particular Lemma 4.6(2) (for compatibility of Gysin maps with pullback, which applies since X → S is smooth), Proposition 4.19 and Proposition 4.13(3) (ensuring that the identification E V * (X) ≃ E Lπ (X) is compatible with base change) and Lemma 5.2 (for compatibility of Euler classes with base change).
Proposition 5.4. Let d be even or d = 1, X/Z[1/d] smooth and proper and V /X a relatively oriented vector bundle. Then for any field k with 2d ∈ k × we have In fact there is a formula n(V k , ρ, HZ) = n a a which holds over any such field, with the coefficients n a ∈ Z independent of k (and zero for all but finitely many a).
Remark 5.5. If d = 1, Proposition 5.4 relies on the novel results about Hermitian K-theory of the integers from [CDH + 20]. In the below proof, this is manifested in the dependence of [BH20, Lemma 3.38(2)] on these results. We will later use Proposition 5.4 for the d = 1 case of Theorem 5.11, whence this result is also using [CDH + 20] in an essential way. For d ≥ 2, the proof is independent of [CDH + 20].
Note that here the assumption that the rank of V equals the dimension of X is included in the hypothesis that V /X a relatively oriented vector bundle (see Definition 1.5).
Proof. Recall the very effective cover functorf 0 and the truncation in the effective homotopy t-structure π eff 0 , for example from [Bac17, § §3,4]. We have a diagram of spectra KO k ←f 0 KO k → π eff 0 KO k ← π eff 0 MSL k ≃ HZ; see [BH17,Example 16.34] for the last equivalence. The functorsf 0 and π eff 0 are lax monoidal in an appropriate sense, so this is a diagram of ring spectra. Moreover all of the ring spectra are strongly SL-oriented (via the ring map MSL k → π eff 0 MSL k ); see also Remark 4.18. Finally all the maps induce isomorphisms on [½, −], essentially by construction. It follows that n(V k , ρ, KO) = n(V k , ρ, HZ) ∈ GW(k). We may thus as well prove the result for n(V k , ρ, KO) instead.
Notation 5.7. Let A ֒→ R and V be a relatively oriented, rank n vector bundle on a smooth proper ndimensional scheme X over A. We have GW(R) ≃ Z ⊕ Z −1 . There are thus unique integers n R , n C ∈ Z such that n(V R , ρ, HZ) = n C + n R 2 + n C − n R 2 −1 .
The integers n R and n C are the Euler numbers of the corresponding real and complex topological vector bundles respectively, at least when X is projective, justifying the notation. To show this, consider the cycle class map CH * (X) → H * (X(C), Z) from the Chow ring of a smooth C-scheme X to the singular cohomology of the complex manifold X(C). See [Ful84,Chapter 19]. Furthermore, there are real cycle class maps from oriented Chow of R-smooth schemes X to the singular cohomology of the real manifold X(R)), discussed in [Jac17] and [HWXZ19] (as well as more refined real cycle class maps defined in [BW20] and [HWXZ19]): For a smooth R-scheme X and a line bundle L → X, consider the real cycle class map CH * (X, L) ∼ = H n (X, K MW n (ω X/k )) ∼ = HZ Lπ (X) → H * (X(C), Z(L)) from the oriented Chow groups twisted by L to the singular homology of the associated local system on X(R). We use results on the real cycle class map due to Hornbostel, Wendt, Xie, and Zibrowius [HWXZ19], including compatibility with pushforwards. Since they only had need of this compatibility in the case of the pushforward by a closed immersion, we first extend this slightly.
Lemma 5.8. Let π : X → Spec k be the structure map of a smooth, projective scheme X of dimension n over the real numbers R. Then the following square commutes, . where the vertical maps are the canonical pushforwards and the horizontal maps are the real cycle class maps.
Proof. π is the composition of a closed immersion i : X ֒→ P n k and the structure map p : P n k → Spec k. The algebraic pushforward π * is the composition p * • i * and the analogous statement holds for the topological pushforward by classical algebraic topology. By [HWXZ19, Theorem 4.7], the pushforward i * commutes with real realization. We may thus reduce to the case where X = P n k is projective n-space over a field k. In this case, π * is an isomorphism by [Fas13, Prop 6.3, Theorem 11.7]. Let s : Spec K → P n k be the closed immersion given by the origin. Since πs = 1, and the real realization maps commute with the algebraic and topological pushforwards of s by [HWXZ19, Theorem 4.7], the real realization maps also commute with the algebraic and topological pushforwards of π.
Proposition 5.9. Let A ֒→ R and V be a relatively oriented, rank n vector bundle on a smooth projective n-dimensional scheme X over A. Then n R and n C are the Euler numbers of the corresponding real and complex topological vector bundles respectively, i.e., n R is the topological Euler number of the relatively oriented topological R n -bundle V (R) associated to the real points of V on the real n-manifold X(R), and n C is the analogous topological Euler number on V (C) → X(C).
Proof. By [HWXZ19, Proposition 6.1], the A 1 -Euler class e(V, HZ) of V R in the oriented Chow group HZ V * (X R ) maps to the topological Euler class of V (R) under the real cycle class map. By Lemma 5.8, it follows that the image of the Euler number n(V, HZ) under the real cycle class map is the topological Euler number of V (R). Under the canonical isomorphism H 0 ( * , Z) ∼ = Z, the real cycle class map GW(R) → H 0 ( * , Z) ∼ = Z is the map taking a bilinear form over R to its signature. It follows that n R is the topological Euler number of V (R). Let , Z) denote the composition of the canonical map to Chow followed by the (usual) cycle class map, and we similarly have Remark 5.10. If proper pushforwards in algebra and topology commute with real realization, as predicted by [HWXZ19, 4.5 Remark], then Proposition 5.9 holds more generally for smooth, proper schemes. It seems that proving this would take us too far afield, however. Alternatively, if V ⊗ R admits a nondegenerate section, then the results of §8 (showing that the Euler number can be computed in terms of Scheja-Storch forms) imply that Proposition 5.9 holds for smooth, proper schemes, arguing as in [SW21, Lemma 5].
Theorem 5.11. Suppose X is smooth and proper over Z[1/2]. Let V be a relatively oriented vector bundle on X and let V k denote the base change of V to k for any field k. Then either where the same formula holds for all fields k of characteristic = 2.
If instead X is smooth proper over Z, then (6) holds for any field k (including fields k of characteristic two).
Recall that for the last claim regarding X smooth and proper over Z, we rely on [CDH + 20]. See Remark 5.5.
Proof. First assume that char(k) = 2. By Corollary 1.4 we have n GS (V k , ρ) = n(V k , ρ, HZ). If the base is Z, then we learn from Proposition 5.4 that there exist a, b ∈ Z (independent of k!) such that n(V k , ρ, HZ) = a + b −1 . which determines a + c and b, so that there only remain at most two possible values for n(V k , ρ, HZ). Now suppose that char(k) = 2 (so that in particular the base is Z). Since 1 = −1 over fields of characteristic 2, we need to show that n GS (V k , ρ) = n C . We may as well assume that k = F 2 . The rank induces an isomorphism GW(F 2 ) ∼ = Z (see e.g. Corollary B.4). Considering the canonical maps in which all but the right-most one are isomorphisms, we get the string of equalities The result follows.
Remark 5.12. The difference of the two values given in Theorem 5.11 is 2 − 1, so the two possibilities can be distinguished by the value of disc n(V k ) in k * /(k * ) 2 for any field k in which 2 is not a square, such as F 3 or F 5 . Such discriminants can be evaluated by a computer, as shown to the second named author by Anton Leyton and Sabrina Pauli.

Refined Euler classes and numbers.
Definition 5.13 (refined Euler class). Let E ∈ SH(S) be a homotopy ring spectrum, X ∈ Sch S , V → X a vector bundle and σ : X → V a section with zero scheme Z = Z(σ). We denote by e(V, σ) = e(V, σ, E) ∈ E V * Z (X) the class corresponding to the composite see Example 4.2.
Remark 5.14. It is clear by construction that refined Euler classes are stable under base change.
Lemma 5.16. The "forgetting support" map E V * Z (X) → E V * (X) sends e(V, σ) to e(V ). Proof. This follows from the commutative diagram Indeed the composite from the bottom left to the top right along the top left represents e(V ), by Definition 5.1 (note that σ is a homotopy inverse to the projection V → X), whereas the composite along the bottom right represents e(V, σ) with support forgotten, by Definition 5.13 and Example 4.2.
Corollary 5.18. Suppose that additionally π : X → S is smooth and proper. Then n(V, σ, ρ) = n(V, ρ) ∈ E 0 (S). Proposition 5.19. Let E ∈ SH(S) be a homotopy ring spectrum, X ∈ Sm S , V a vector bundle over X and σ a section of V . Then e(V, σ, E) = σ * z * (1), where z : X → V is the zero section and we use the canonical isomorphism N z ≃ V to form the pushforward z * .
Proof. Let p : V → X be the projection and s 0 : V → p * V the canonical section. Then σ * (p * V, σ 0 ) = (V, s) and hence, using Remark 5.14, it suffices to show that z * (1) = e(p * V, σ 0 , E). By [EHK + 20a, last sentence of §2.1.1] we know that is the purity equivalence. In the case of the zero section of a vector bundle, it just takes the tautological form , and hence indeed sends 1 to e(p * V, σ 0 ).
It follows from the above that Euler classes of vector bundles are determined by Euler classes of vector bundles with non-degenerate sections.
Example 5.20. Let V → X be a vector bundle and σ a section. Suppose 1/2 ∈ S. The Euler Class of V in KW = KO[η −1 ] is given by the Koszul complex with its canonical symmetric bilinear form. Indeed we may assume that V has a regular section, in which case this follows from the existence of the morphism of cohomology theories BL naive → KW and Proposition 2.11.
We deduce that the "Meta-Theorem" of §3.4 holds in this setting, even in a slightly stronger form with supports: Corollary 5.21. If σ is a non-degenerate section (i.e. locally given by a regular sequence), then i * (1) = e(V, σ) ∈ E V * Z (X). In particular, forgetting supports (taking the image along E V * Z (X) → E V * (X)) we have i * (1) = e(V ) ∈ E V * (X), i.e. Theorem 3.9 holds in this situation.
Proof. The second statement follows from the first and Lemma 5.16; hence we shall prove the first. Consider the following cartesian square Z A well-known consequence of regularity of σ is that this square is tor-independent. 8 Since i * is compatible with tor-independent base change (Lemma 4.6(2)), we deduce from Proposition 5.19 that e(V, σ, E) = σ * z * (1) = i * i * (1) = i * (1).
This was to be shown.
It follows that, as explained in §3.4, both the ordinary and refined Euler numbers in E-cohomology can be computed as sums of local indices. The remainder of this paper is mainly concerned with determining these indices, for certain examples of E.
Remark 5.22. These results can be generalized slightly. Fix a scheme S and an SL-oriented ring spectrum E ∈ SH(S).
(1) Let X/S smoothable lci, V /X a vector bundle, relatively oriented in the sense of Definition 1.5. If X/S is in addition proper, then we can transfer the Euler class along the structure morphism (see §4.2.4) as before to obtain an Euler number n(V, ρ, E) ∈ E 0 (S). More generally, without assuming X/S proper, given a section σ with zero scheme Z proper over S, we obtain the refined Euler number n(V, σ, ρ, E).
8 To see this note that z * O X can be resolved by the Koszul complex K(p * V, σ 0 ) for the tautological section σ 0 of the pullback p * V of V along p : V → X. It follows now from the projection formula that z * O X ⊗ L σ * O X ≃ σ * σ * z * O X ≃ σ * σ * K(p * V, σ 0 ) ≃ σ * K(V, σ). Since by definition σ locally corresponds to a regular sequence (and σ is affine), the claim follows.
(2) Let X/S be arbitrary, V a vector bundle, σ a section of V with zero scheme i : Z → X, and suppose that i is a regular immersion (but σ need not be a non-degenerate section, i.e. Z could have higher than expected dimension). In this case there is an excess bundle E = cok(N Z/X → V | Z ). 9 A straightforward adaptation of the proof of Corollary 5.21, using the excess intersection formula [DJK18, Proposition 3.3.4], shows that e(V, σ, E) = i * (e(E, E)).
(3) Putting everything together, let X/S be proper smoothable lci, V a relatively oriented vector bundle and σ a section with zero scheme Z regularly immersed in X. Then Here the sum is over clopen components Z ′ of Z and ρ ′ denotes the induced relative orientation of E. Note that if σ is non-degenerate on Z ′ , i.e. E| Z ′ = 0, then e(E| Z ′ , E) = 1 ∈ E 0 (Z ′ ) and n(E| Z ′ , ρ ′ , E) = ind Z ′ (σ) as before.
6. d-Dimensional planes on complete intersections in projective space Remark 6.1. Suppose X is a smooth, proper Z-scheme with geometrically connected fibers and PicX torsion free, for example, X a Grassmannian or projective space. For a relatively orientable vector bundle V → X defined over Z, there are at most two isomorphism classes of relative orientations. Namely, by assumption, there is a line bundle L → X and isomorphism ρ : L ⊗2 ∼ = −→ ω X/Z ⊗ det V . Since PicX is torsion free, any relative orientation is an isomorphism L ⊗2 ∼ = −→ ω X/Z ⊗ det V , whence two such differ by a global section of Hom(L ⊗2 , L ⊗2 ) ∼ = O(X) * . By hypothesis on X, the fibers of the pushforward of O X all have rank 1, whence this pushforward is O Z and O(X) * ∼ = Z * = {±1}. Thus any relative orientation is isomorphic to ρ or −ρ. We then have n(V, ρ) = −1 n(V, −ρ).
Consequently we suppress the choice of orientation and just write n(V ). Beware that this does not mean that every vector bundle is relatively orientable, though! Remark 6.2. There is a canonical relative orientation for certain classes of vector bundles on Grassmannians X = Gr(d, n): a point p of X with residue field L corresponds to a dimension d + 1 subspace of L n+1 . A choice of basis {e 0 , . . . , e n } of L n+1 such that the span of {e 0 , . . . , e d } is p defines canonical local coordinates for an affine chart isomorphic to A (n−d)(d+1) with p as the origin (see for example [KW21,Definition 42]). This defines local trivializations of the tautological and quotient bundles on X, and therefore also of their tensor, symmetric, exterior powers and their duals. A vector bundle V formed from such operations on the tautological and quotient bundles and which is relatively orientable on X inherits a canonical relative orientation ρ such that the local coordinates and trivializations just described are compatible with ρ in the sense of [KW21, Definition 21]. This is described in [KW21, Proposition 45] in a special case, but the argument holds in the stated generality. (One only needs the determinants of the clutching functions to be squares which follows from the relative orientability of V . Together with the explicit coordinates this gives the relative orienation.) This relative orientation has the property that it is defined over Z and for any very non-degenerate section σ, the data just described gives a system of coordinates in the sense of Definition 2.28.
Corollary 6.3. Let d ≤ n be positive integers, and let X = Gr(d, n) be the Grassmannian of d-planes P d in P n . Let V = ⊕ j i=1 Sym ni S * , where S denotes the tautological bundle on X, and n 1 , . . . , n j are positive integers such that rank V = dim X and V is relatively orientable, i.e. such that ni d+1 ni+d d + n + 1 is even. Then over any ring in which 2 is invertible (where we interpret n(V ) as n(V, KO)), or any field (where we interpret n(V ) as n GS (V )). ). Thus V is relatively orientable as a bundle over Z and Remarks 6.1 and 6.2 apply.
By the Z[1/2] case of Theorem 5.11, it is enough to show that the discriminant of n(V Fp ) is trivial for some prime p congruent to 1 mod 4, and such that 2 is not a square. See Remark 5.12. Let EM(W) denote the Eilenberg-MacLane spectrum of the η-inverted Milnor-Witt sheaves K MW * [η −1 ], cf. [Lev19, Remark 3.1], and consider the associated Euler class e(V, EM(W)). Then n(V, EM(W)) determines the Witt class of n(V ), and hence the discriminant of n(V ) as well.
d is even: Suppose that d is even. Let π denote the structure map of X. Since n(V, EM(W)) = π * j i=1 e(Sym ni S * , EM(W)), it is enough to show that e(Sym n1 S * , EM(W)) = 0. By the Jouanolou device, we may assume any vector bundle is pulled back from the universal bundle. It is therefore enough to show the same for the dual tautological bundle on the universal Grassmannian BGL d+1 , i.e., let S * d+1 denote the dual of the tautological bundle on BGL d+1 ; we show that e(Sym n1 S * d+1 , EM(W)) = 0. By Ananyevskiy's splitting principle [Ana15, Theorem 6] and its extension due to M. Levine [Lev19,Theorem 4.1], we may show the vanishing of the EM(W)-Euler class of Sym n1 S * d+1 after pullback to BSL 2 × BSL 2 × . . . × BSL 2 × BSL 1 via the map classifying the external Whitney sum of the tautological bundles. Here we use that d + 1 is odd. This pullback of Sym n1 S * d+1 contains the odd-rank summand Sym n1 S * 1 , and therefore its EM(W)-Euler class is 0 as desired [Ana15, Lemma 3] [Lev19, Lemma 4.3].
d is odd: Let k be a finite field whose order is prime to 2 j i=1 (n i )!, congruent to 1 mod 4 (so −1 is a square), and such that 2 is not a square. By Theorem 5.11, it suffices to show that the discriminant of n(V, Gr(d, n)) ∈ W(k) ∼ = GW(k)/Zh is trivial, cf. Remark 5.12.
Define r in Z so that d = 2r + 1. Let S * d+1 denote the dual tautological bundle on BSL d+1 and let p 1 , . . . , p r , p r+1 and e in EM(W) * (BSL d+1 ) denote its Pontryagin and Euler classes respectively. (Often one would let p i be the Pontryagin classes of the tautological bundle, not its dual, but this is more convenient here.) By Lemma 6.5, e(Sym ni S * d+1 , EM(W)) is in the image of Z[p 1 , . . . , p r , e] → EM(W) * (BSL d+1 ). (Note that we have omitted p r+1 as p r+1 = e 2 [Ana15, Corollary 3].) Therefore e(V, Gr(d, n)) can be expressed as a polynomial with integer coefficients in the Pontryagin classes and Euler class of the dual tautological bundle S * on Gr(d, n). By Lemma 6.7, it follows that disc n(V, Gr(d, n)) = 1 in k * /(k * ) 2 as desired.
M. Levine [Lev19] uses the normalizer N of the standard torus of SL 2 and bundlesÕ(a) andÕ − (a) for a in Z corresponding to the representations respectively, to compute characteristic classes, and we use his technique. We will use the notatioñ O (−) (a) to mean eitherÕ(a) andÕ − (a) when a claim holds for both possibilities. We likewise use the EM(W)-Pontryagin (or Borel) classes of a vector bundle with trivialized determinant of Panin and Walter [PW10b]. See [Ana15, Introduction, Section 3] or [Lev19, Section 3] [Wen20, Section 2] for background on these classes. Let e i ∈ EM(W) * (N r+1 ) denote the pullback of e(S * 2 , EM(W)) under the ith projection BN r+1 → BN composed with the canonical map BN → BSL 2 .
Given vector bundles V and E on schemes X and Y , respectively, let V ⊠ E denote the vector bundle on X × Y given by the tensor product of the pullbacks of V and E.
Let p 1 , . . . , p r and e in EM(W) * (BSL d+1 ) denote the Pontryagin and Euler classes respectively of the dual tautological bundle S * d+1 on BSL d+1 . Lemma 6.5. Let d = 2r+1 be an odd integer and let n be a positive integer. Let k be a field of characteristic not dividing 2n!. Then e(Sym n S * d+1 , EM(W)) is in the image of Z[p 1 , . . . , p r , e] → EM(W) * (BSL d+1 ). Proof. Let f denote the composite of the (r+1)-fold product of the canonical map BN → BSL 2 with the map BSL r+1 By inspection, the symmetric powers of the tautological bundleÕ(1) on BN split into a sum of bundles of rank ≤ 2, cf. [Lev19, p. 38]: , when a = 2b + 1 is odd when a = 2b is even and b is even where γ is the line bundle corresponding to the representation N → GL 1 sending the torus to 1 and 0 1 −1 0 to −1.
Combining Equations (14) and (15), we can decompose f * Sym n S * d+1 into a direct sum with summands which have various numbers of factors of rank 2. Separate these summands into those with at least two rank 2 factors and those with only one rank 2 factor, if any of the latter sort appear. (This occurs when we can take all but one a i to be even.) The direct sum of the latter such terms can alternatively be expressed as a sum of pullbacks of Sym aiÕ (1) under some projection N r+1 → N tensored with some γ's pulled back from other projections. We may ignore the factors of γ by [Lev20, §10 p. 78 (2)] because e(γ) = 0 as γ is a bundle of odd rank. SinceÕ is rank 2, and the characteristic of k does not divide 2a i , we may apply [Lev19, Theorem 8.1] and conclude that e(Sym aiÕ (1)) is an integer multiple of a power of e(Õ(1)). Since the summands are symmetric under the permutation action of the symmetric group on r + 1 letters on BN r+1 , it follows that the Euler class of these summands is an integer multiple of a power of e.
We now consider the Euler class of the rest of the summands. Namely, it suffices to show that the Euler class ǫ 1 of the summands with at least two rank 2 factors is also in the image of Z[p 1 , . . . , p r , e]. We may again ignore the factors of γ, as these do not change the Euler class. By Lemma 6.4, ǫ 1 is the image of an element of Z[e 2 1 , . . . , e 2 r+1 ]. Moreover, because each tuple (a 1 , . . . , a r+1 ) of the direct sum occurs in every permutation, we may choose an element of Z[e 2 1 , . . . , e 2 r+1 ] which is invariant under the permutation action of the symmetric group on r + 1 letters and which maps to ǫ 1 . Thus, ǫ 1 is in the image of the map Z[σ 1 (e 2 1 , . . . , e 2 r+1 ), . . . , σ r+1 (e 2 1 , . . . , e 2 r+1 )] → EM(W) * (BN r+1 ), where σ i denotes the ith elementary symmetric polynomial. Since σ i ((e 2 1 , . . . , e 2 r+1 )) is the pullback to BN of p i (S * d+1 → BSL d+1 , EM(W)), we have that ǫ 1 is in the image of Z[p 1 , . . . , p r , e] as desired.
Lemma 6.6. Let d = 2r+1 be odd. For any non-negative integers a 1 , . . . , a r+1 , b such that (4i)a i +b(d+ 1) = (d + 1)(n − d) and b ≡ n + 1 mod 2, the monomial , or in other words, there exists c in Z such that e b r+1 i=1 p ai i = ce n−d . Proof. Let Q denote the quotient bundle on Gr(d, n), defined by the short exact sequence In particular, the rank of Q is n − d. The non-vanishing Pontryagin classes of S d+1 are p 1 , . . . , p r , p r+1 with e 2 = p r+1 . Define s so that n − d = 2s or n − d = 2s + 1 depending on whether n − d is odd or even. Let p ⊥ 1 , . . . , p ⊥ s denote the non-vanishing Pontryagin classes of the dual to the quotient bundle Q * on Gr(d, n). By [Ana15, Lemma 15], (Gr(d, n)). Setting the notation A for the ring where H * (C Gr(r, r + s); Z) denotes the singular cohomology of the C-manifold associated to the C-points of the Grassmannian Gr(r, r + s), sending the ith Chern class of the dual tautological bundle to p i . The top dimensional singular cohomology H (r+1)s (C Gr(r, r + s); Z) is isomorphic to Z by Poincaré duality. Under our chosen isomorphism, the monomial p s r+1 corresponds to the top Chern class c (r+1)s (S ⊕s r+1 → C Gr(r, r + s)) of the direct sum of s-copies of the dual tautological bundle, which is a generator (with the usual C-orientations, c (r+1)s (S ⊕s r+1 → C Gr(r, r + s)) counts the number of linear subspaces of dimension r in a complete intersection of s linear hypersurfaces in CP r+s , and this number is 1, cf. Remark 6.9 and Lemma 6.7). Therefore, for any monomial in H * (C Gr(r, r + s); Z).
Since d is odd, n + 1 ≡ n − d mod 2, and therefore b ≡ n − d mod 2. Note that if n − d is odd, b ≥ 0. We may then define a non-negative integer b ′ by the rule By Equation (16), there is an integer c so that we have the equality in H * (C Gr(r, r + s); Z). Applying τ , we see that which implies the claim, either immediately in the case that n − d is even, or by multiplying by e if n − d is odd.
Lemma 6.7. Let d be odd. Suppose k is a finite field such that −1 is a square. For any non-negative integers a 1 , . . . , a r+1 , b such that b(d + 1) + r+1 i=1 4ia i = (d + 1)(n − d) and b ≡ n + 1 mod 2, the pushforward π * (e b r+1 i=1 p ai i ) has trivial discriminant. Remark 6.8. The condition b ≡ n + 1 mod 2 ensures that e b r+1 i=1 p ai i lies in the appropriate twist of the Witt-cohomology of Gr(d, n), i.e. e b r+1 i=1 p ai i is in EM(W) * (Gr(d, n), ω Gr(d,n) /k), as opposed to EM(W) * (Gr(d, n)), so that we may apply π * . The condition on the sum b(d + 1) + r+1 i=1 4ia i ensures that e b r+1 i=1 p ai i lies in the (d + 1)(n − d)-degree EM(W) * -cohomology of Gr(d, n), so the codomain of π * is W(k).
Proof. By Lemma 6.6, it suffices to show that disc π * e n−d = 1. We may identify π * e n−d with the Euler number of ⊕ n−d j=1 S * , π * e n−d = n(⊕ n−d j=1 S * , EM(W)). Let x 0 , . . . , x n be coordinates on projective space P n k = Proj k[x 0 , . . . , x n ]. The Euler number n(⊕ n−d j=1 S * , EM(W)) can be calculated with the section σ = ⊕ n i=d+1 x i as in §2.4. There is an analogous section The zero locus σ Z = 0 is the single Z-point of the Grassmannian associated to the linear subspace of P n k given by x d+1 = x d+2 = . . . = x n = 0. The vanishing locus of σ Z is the origin of the affine space A 1 0 · · · 0 a 0,d+1 a 0,d+2 · · · a 0,n 0 1 · · · 0 a 1,d+1 a 1,d+2 · · · a 1,n . . . . Namely, x u ((a ij )) = d l=0 (x u (a ij ))(ẽ l )x l for u = d + 1, . . . , n. As a subscheme, σ Z = 0 is therefore the zero locus of (x u (a ij ))(ẽ l ) = a l,u for l = 0, . . . , d and u = d + 1, . . . , n. Thus the subscheme of Gr(d, n) Z given by {σ Z = 0} is a section of the structure map Gr(d, n) Z → Spec Z. In particular, it is finite andétale of rank 1. It follows that the Jacobian of σ (which is described further at the beginning of Section 6.2) ) is nowhere vanishing. Thus under the relative orientation , we have that Jacσ is a nowhere vanishing section of the restriction of L ⊗2 . Thus Jacσ k is either −1 or 1 ; but −1 = 1 by assumption. Since n(⊕ n−d j=1 S * k , EM(W)) = Jacσ k (cf. Example 2.21), this proves the claim.
ni+d d or empty, with the empty case occurring exactly when one or both of (d + 1)(n − d) − rank V and n − 2r − j is less than 0. In particular, when (d + 1)(n − d) − rank V = 0, the zeros of σ are isolated andétale over k for a general complete intersection X. The canonical relative orientation (Remark 6.2) of V determines an isomorphism Hom(det T Gr(d, n), det V ) ∼ = L ⊗2 for a line bundle L on Gr(d, n). The Jacobian determinant Jacσ at a zero p of σ is an element of the fiber of the vector bundle Hom(det T Gr(d, n), det V ) at p. Choosing any local trivialization of L, we have a well-defined element JacΣ(p) in k(p)/(k(p) * ) 2 , which can also be computed by choosing a local trivialization of V and local coordinates of Gr(d, n) compatible with the relative orientation and computing JacΣ(p) = det ∂σ k ∂x l .
Corollary 6.9. Let X = {F 1 = F 2 = . . . = F j = 0} ⊂ P n be a general complete intersection of hypersurfaces F i = 0 of degree n i in P n k projective space over a field k. Suppose that where • k(PL) denotes the residue field of PL viewed as a point on the Grassmannian • n C (respectively n R ) is the topological Euler number of the complex (respectively real) vector bundle associated to the algebraic vector bundle V = ⊕ j i=1 Sym ni S * given the canonical relative orientation (6.2) • and Jacσ is the Jacobian determinant.
Proof. By [DM98, Théorème 2.1], the zeros of σ are isolated andétale over k. It follows [KW21, p.18, Proposition 34] that for a zero of σ corresponding to the d-plane PL, the local index is computed ind PH PL σ = tr k(L)/k Jac σ(PL) . See Section 2.4 for the definition of the notation ind PH . Thus n PH (V, σ) = d-planes PL in X tr k(L)/k Jac σ(PL) by Definition 2.24. Corollary 6.3 computes n PH (V, σ). Proposition 5.9 shows that n R and n C are the claimed topological Euler numbers.
Remark 6.10. Note that Corollary 6.9 is a weighted count of the dimension d hyperplanes on the complete intersection X, depending only on n 1 , . . . , n j and not on the choice of polynomials F 1 , . . . , F j as long as these are chosen generally.
Example 6.11. Examples where Corollary 6.9 applies include: i) lines on a degree 2n − 1 hypersurface of dimension n, ii) 3-planes on a degree d hypersurface of dimension 2 + 1 3 d+3 3 , when this is an integer. iii) Lines on a complete intersection of two degree n − 2 polynomials in P n for n odd. Matthias Wendt's oriented Schubert calculus shows that enriched intersections of Schubert varieties are determined by the R and C realizations in the same manner [Wen20, Theorem 8.6], as well as giving enumerative applications [Wen20, Section 9].
For d = j = 1 and n 1 = 3, Corollary 6.3 is work of Kass and the second named author [KW21] over a general field. For d = 1 and general j and n i , it is work of M. Levine [Lev19] over a perfect field either of characteristic 0 or of characteristic prime to 2 and the odd n i s. Our result eliminates the assumption on the characteristic, and generalizes to arbitrary relatively orientable d, n, n 1 , . . . , n j .
In order to obtain an enumerative theorem whose statement is independent of A 1 -homotopy theory, one needs an arithmetic-geometric interpretation of the local indices: Question 6.12. Can the local indices ind PH PL σ = tr k(L)/k Jac σ(PL) be expressed in terms of the arithmetic-geometry of the d-plane PL on X?
Such expressions are available over R for d = j = 1 [FK21], and over a field k of characteristic not 2, for lines on a cubic surface [KW21], lines on a quintic 3-fold [Pau20], and points on a complete intersection of hypersurfaces [McK21]. S. Pauli has interesting observations on such results for lines on the complete intersection of two cubics in P 5 . Dropping the assumption that the zeros of σ are isolated, she can compute contributions from infinite families of lines on a quintic 3-fold in some cases [Pau20]. An alternative point of view in terms of (S)pin structures for d = j = 1 and n 1 = 5, as well as computations of the real Euler number is discussed [Sol06, Example 1.6, Theorem 8.8].
Example 6.13. The computations of the Euler classes of C and R-points in Finashin and Kharlamov's paper [FK15,p. 190] imply the following enriched counts of 3-planes on hypersurfaces over any field k. n(Sym 3 S * → Gr(3, 8)) = 160839 1 + 160650 −1 corresponds to an enriched count of 3-planes in a 7-dimensional cubic hypersurface. Namely, for a general degree 3 polynomial F in 9 variables, the corresponding cubic hypersurface X ⊂ P 8 contains finitely many 3-planes as discussed above and 3−planes P ⊂X tr k(L)/k Jac σ F (P ) = 160839 1 + 160650 −1 , where σ F is the section of Sym 3 S * defined by σ F [P ] = F | P .
7. Indices of sections of vector bundles and A 1 -degrees 7.1. A 1 -degrees. Recall the following.
Definition 7.1 (local A 1 -degree). Let S be a scheme, X ∈ Sch S and F : X → A n S be a morphism. We say that F has an isolated zero at Z ⊂ X if Z is a clopen subscheme of Z(F ) such that Z/S is finite. Now let X ⊂ A n S be open and suppose that F has an isolated zero at Z ⊂ X. We define the local as the morphism corresponding to the unstable map Here we use that by assumption Z(F ) ≃ Z ∐ Z ′ , and hence X/X \ Z(F ) ≃ X/X \ Z ∐ X/X \ Z ′ .
Example 7.2. If S = Spec(k) is the spectrum of a field, then an isolated zero z ∈ A n k of F : A n k → A n k in the usual sense is also an isolated zero {z} ⊂ A n k in the above sense, and deg z (F ) ∈ GW(k) is the usual local A 1 -degree of [KW19, Definition 11].
Lemma 7.3. Let X, Y ∈ Sm S and (Z, U, φ, g) be an equationally framed correspondence from X to Y [EHK + 17, Definition 2.1.2]; in other words Z ⊂ A n X , U is anétale neighbourhood of Z, g : U → Y and φ : U → A n is a framing of Z. Then the following two morphisms are stably homotopic T n ∧ X + ≃ (P 1 ) ∧n ∧ X + → (P 1 ) ×n X /(P 1 ) ×n Proof. For E ∈ SH(S), precomposition with the first morphism (desuspended by T n ) induces a map E(Y ) → E(X) known as the Voevodsky transfer. Precomposition with the second map induces an "alternative Veovedsky transfer". It suffices (by the Yoneda lemma) to show that these transfer maps have the same effect (even just on π 0 ), for every E. In [EHK + 20a, Theorem 3.2.11], it is shown that the Voevodsky transfer coincides with a further construction known as the fundamental transfer. In that proof, all occurrences of (P 1 ) ∧n can be replaced by P n /P n−1 ; one deduces that the alternative Voevodsky transfer also coincides with the fundamental transfer. The result follows.
is the same as the endomorphism given by the equationally framed correspondence defined by F .
Proof. By definition, deg Z (F ) is given by the second morphism of Lemma 7.3, whereas the equationally framed correspondence is given by the first morphism. The result follows.
Recall also the following. where ̟ : Z → S is the projection. The endomorphism of ½ S corresponding to α is given by ̟ * (1). 7.2. Main result. Let X/S be smooth, V /X a relatively oriented vector bundle with very non-degenerate section σ and zero scheme Z (which is thus finite over S). Let Z ′ ⊂ Z be a clopen component and suppose there are coordinates (ψ, ϕ, σ ′ ) for (V, X, σ, ρ, Z ′ ) as in Definition 2.28. Then σ ′ = (F 1 , . . . , F d ) determines a function F : A d S → A d S , and ϕ(Z ′ ) is an isolated zero of F . Theorem 7.6. Assumptions and notations as above. Let E ∈ SH(S) be an SL-oriented ring spectrum with unit map u : ½ → E. Then Proof. By Corollary 7.4, Proposition 7.5 and §4.2 The result now follows from Proposition 3.13. Example 7.7. Suppose S = Spec(k) is a field. Then for well-chosen E (e.g. E = KO or E = HZ), the unit map is an isomorphism. We deduce that ind z (V, σ, ρ, E) is essentially the same as deg ϕ(z) (F ).

Euler numbers in KO-theory and applications
As explained in §3.3 and §5.3, we have the motivic ring spectrum KO related to Hermitian Ktheory made homotopy invariant, and associated theories of Euler classes and Euler numbers. Using (for example) the construction in §A, we can define KO even if 1/2 ∈ S. There is still a map GW(S) → KO(S), however we do not know if this is an equivalence, even if S is regular (but we do know this if S is regular and 1/2 ∈ S). Recall that for any lci morphism f we put ω f = detL f and ω f = det L f . We show in Proposition B.1 that for f : X → Y an lci morphism, we have f ! (O) ≃ ω f . Via Proposition B.1, coherent duality (i.e. the adjunction f * ⊣ f ! ) thus supplies us with a canonical trace map provided that f is also proper. One expect that this can be used to build a map f * : GW(X, f ! L) → GW(Y, L), and moreover that the following diagram commutes If we assume that 1/2 ∈ S, replace GW by W and KO by KW, then mapsf * can be defined and studied using the ideas from §2.2; see also [CH11]. Levine-Raksit [LR20] show that the (modified) diagram commutes provided that X and Y are smooth over a common base with 1/2 ∈ S.
If instead we assume that f is finite syntomic, then the analogous result is proved (for GW → KO and without 1/2 ∈ S) in Corollary A.4. This is the only case that we shall use in the rest of this section. Recall the construction of the Scheja-Storch form −|− SS from Definition 2.20.
Corollary 8.2. Let X ∈ Sm S , V /X a relatively oriented vector bundle with a very non-degenerate section σ, and Z a clopen component of the zero scheme Z(σ). Suppose there exists coordinates (ψ, ϕ, σ ′ ) around Z, as in Definition 2.28. Then Proof. By Proposition 3.13, we may assume that ψ = id and so on; so in particular X ⊂ A n S . The result now follows from the identification of the transfers in Corollary A.4 (telling us that the index is given by the trace form from coherent duality) and Theorem 2.18 (identifying the coherent duality form with the Scheja-Storch form).
Corollary 8.3. Let S be regular semilocal scheme over a field k of characteristic = 2.
(1) Let ̟ : S ′ → S be a finite syntomic morphism, and τ a trivialization of L ̟ ∈ K(S ′ ). Then the associated endomorphism of the sphere spectrum over S is given under the isomorphism   2) is a special case of (1). This concludes the proof.
Then ̟ * (1) is given by the image in GW (X) of the symmetric bilinear form on Proof. Using unramifiedness of GW [Mor05, Lemma 6.4.4], we may assume that X is the spectrum of a field. Then X ′ is semilocal, so L ≃ O and we obtain (up to choosing such an isomorphism) ω X ′ /X ρ ′

≃ O.
Since L X ′ /X has constant rank (namely 0), it follows from [BH98, Lemma 1.4.4] that L X ′ /X ≃ 0 ∈ K(X ′ ). The set of homotopy classes of such trivializations is given by K 1 (X ′ ), which maps surjectively (via the determinant) onto O × (X ′ ). It follows that there exists a trivialization τ : 0 ≃ L X ′ /X ∈ K(X ′ ) such that det(τ ) = ρ ′ . Hence we have a commutative diagram It follows from Corollary 8.3 that the bottom row is the form ̟ * (1) arising from the orientation ρ; by what we just said this is the same as the top row, which is the form we were supposed to obtain.
This concludes the proof.
We also point out the following variant.
Corollary 8.5. Let l/k be a finite extension of fields, 1/2 ∈ k. Them Morel's absolute transfer [Mor12, §5.1] tr l/k : GW (l, ω l/k ) → GW (k) is given as follows. Let φ : V ⊗ l V → l be an element of GW (l), α ∈ ω × l/k . Then where η l/k : ω l/k → k is the (k-linear) trace map of coherent duality (see §B.1). Remark 8.7. We expect that all of the results in this section extend to fields of characteristic 2 as well. This should be automatic as soon as KO is shown to represent GW in this situation (over regular bases, say).

Appendix A. KO via framed correspondences
In this section we will construct a strong orientation on KO, and identify some of the transfers. We would like to thank M. Hoyois for communicating these results to us. For another approach to parts of the results in this section see [LÁ17].
We shall make use of the technology of framed correspondences [EHK + 17]. We write Corr fr (S) for the symmetric monoidal ∞-category of smooth S-schemes and tangentially framed correspondences. Denote by FSyn or ∈ CAlg(P Σ (Corr fr (S))) the stack of finite syntomic schemes Y /X together with a choice of trivialization det L Y /X ≃ O, with its standard structure of framed transfers. This is constructed in [EHK + 20b, Example 3.3.4].
Write Bil ∈ CAlg(P Σ (Sch S )) for the presheaf sending X to the 1-groupoid of pairs (V, φ) with V → X a vector bundle and φ : V ⊗ V → O X a non-degenerate, symmetric bilinear form. The commutative monoid structure is given by ⊗. If f : X → Y is a finite syntomic morphism then a choice of trivialization detL f ≃ ω f ≃ O X induces an additive mapf * : Bil ≃ (X) → Bil ≃ (Y ); see e.g. §2.2.
The morphism FSyn or → Bil is informally described as follows: a pair (f : X → Y finite syntomic, Proof. Denote by K • ∈ P Σ (Sm S ) the rank 0 part of the K-theory presheaf and by Corr fr ((Sm S ) /K • ) the subcategory of the category constructed in [EHK + 20b, §B] on objects (X, ξ) with X ∈ Sm S , ξ of rank 0, and morphisms those spans whose left leg is finite syntomic. There are symmetric monoidal functors γ : (Sm S ) /K • → Corr fr ((Sm S ) /K • ) and δ : Corr fr (S) → Corr fr ((Sm S ) /K • ).
We first lift Bil to P Σ (Corr fr ((Sm S ) /K • )); we let Bil(X, ξ) be the 1-groupoid of vector bundles with a symmetric bilinear form for the duality Hom(−, det ξ). Since Bil is 1-truncated, we only need to specify a finite amount of coherence homotopies, so this can be done by hand. Since δ is symmetric monoidal δ * is lax symmetric monoidal and hence δ * (Bil) produces the desired lift.
Proof. The spectrum (Σ ∞ fr GW)[β −1 ] ∈ SH fr (S) can be modeled by the framed T 4 -prespectrum (GW, GW, . . . ) with the bonding maps given by multiplication by β. Under the equivalence SH fr (S) ≃ SH(S) [Hoy21], this corresponds to the same prespectrum with transfers forgotten. This is KO by definition.
In particular we have constructed an E ∞ -structure on KO.
Recall that by our conventions, X and Y are separated and of finite type over some noetherian scheme S; in particular they are themselves noetherian. We strongly believe that these assumptions are immaterial.
Proof. Both sides are compatible with passage to open subschemes of X. Locally on X, f factors as a regular immersion followed by a smooth morphism, say f = pi. For either a regular immersion or a smooth morphism g, we have isomorphisms Similarly we have a canonical cofiber sequence i * L p → L pi → L i and hence (20) detL pi ≃ i * detL p ⊗ detL i .
Combining (18), (19) and (20), we thus obtain an isomorphism We have thus shown that f ! O Y is locally isomorphic to detL f (via α p,i ), and hence that A := f ! O Y ⊗ ( detL f ) −1 is an O X -module concentrated in degree 0. Exhibiting an isomorphism as claimed is the same as exhibiting A ≃ O X , or equivalently a section a ∈ Γ(X, A) which locally on X corresponds to an isomorphism. Since A is 0-truncated, we may construct a locally. In other words, we need to exhibit a cover {U n } n of X and isomorphisms α n : f ! O Y | Un ≃ detL f | Un such that on U n ∩ U m we have α n ≃ α m . Hence, since we are claiming to exhibit a canonical isomorphism, we may do so locally on X. We may thus assume that f factors as pi, for a smooth morphism p : V → Y and a regular immersion i : X → V . We have already found an isomorphism in this situation, namely α p,i . What remains to do is to show that this isomorphism is independent of the choice of factorization f = pi.
Thus let i ′ : X → V ′ and p ′ : V ′ → Y be another such factorization. We need to show that α p,i = α p ′ ,i ′ . By considering V ′ × Y V , we may assume given a commutative diagram where q is smooth. If in (20) both f and g are smooth, then the isomorphism arises from the first fundamental exact sequence of Kähler differentials [Har66, Proposition III.2.2], and hence is the same as the isomorphism (19). It follows that we may assume that p = id. The isomorphism i ′! q ! ≃ i ! is explained in [Har66, Proposition III.8.2] and reduces via formal considerations (that apply in the same way to detL − ) to the case of a smooth morphism with a section.
We are thus reduced to the following problem. Let p : V → X be smooth and i : X → V a regular immersion which is a section of p. The coherent duality formalism provides us with an isomorphism ω X/V ⊗ i * ω V /X ≃ O X ; we need to check that this is the same as the isomorphism det L i ⊗ i * det L p ≃ det L id = O X coming from (20). By [Har66, Lemma III.8.1, Definition III.1.5], the first isomorphism arises from the second fundamental exact sequence of Kähler differentials. This is the same as the second isomorphism.
This concludes the proof.